68 Proceedings of the Royal Irish Academy. 



Therefore, 



<fi u V w =0. 



X \ IX. V 



y y )^' v 



% X" ix" v" 

 Therefore the equation of the cubic having «<, e;, ^t', as polar conies is 



= 0. 



^ U V w 

 X X fji V 



y A' i^: v' 



% \" fj." v" 



and u, V, w, are the polar conies of the points whose coordinates are 

 (X, X', X"), {fx, fx', [x"), {v, v', v"), respectively, since V, V, W are 

 the polar conies of $ with respect to the points (1, 0, 0), (0, 1, 0), 

 (0, 0, 1). 



The equations (1) show that there are three points, say P, Q, H, 

 associated with any three conies u, v, w, such that the polar line of R 

 with respect to v, and Q with respect to w, of P with respect to w, 

 R with respect to u ; and Q with respect to m, and P with respect to 

 V, are coincident. If P, Q, and R are not coUinear, it is possible to find 

 a cubic having m, v, w as polar conies, and they are the polar conies of 

 the points P, Q, and R. 



The algebraical statement of the problem under consideration is to 

 transform thi-ee ternary quadrics, so that they may be the first deriveds 

 of a ternary cubic. The corresponding problem for binary quantics is 

 to transform two binary cubics so that they may be the first deriveds 

 of a binary quantic ; and I would remark, in conclusion, that the above 

 method of investigation readily gives a solution of this latter problem 

 also. 



{ 



